Dynamics of Schwarz reflections: the mating phenomena

Abstract

We initiate the exploration of a new class of anti-holomorphic dynamical systems generated by Schwarz reflection maps associated with quadrature domains. More precisely, we study Schwarz reflection with respect to the deltoid, and Schwarz reflections with respect to the cardioid and a family of circumscribing circles. We describe the dynamical planes of the maps in question, and show that in many cases, they arise as unique conformal matings of quadratic anti-holomorphic polynomials and the ideal triangle group.

Same as accepted version, to appear in "Ann. Sci. Éc. Norm. Supér. (4)"

References in corpus (2)

• Normal random matrix ensemble as a growth problem
• On a Family of Rational Perturbations of the Doubling Map

Cited by in corpus (10)

• Circle packings, kissing reflection groups and critically fixed anti-rational maps
• Univalent Polynomials and Hubbard Trees
• On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
• Schwarz reflections and the Tricorn
• Schwarz reflections and anti-holomorphic correspondences
• Bers Slices in Families of Univalent Maps
• Combining Rational maps and Kleinian groups via orbit equivalence
• The Sullivan dictionary and Bowen-Series maps
• Antiholomorphic correspondences and mating I: realization theorems
• Matings, holomorphic correspondences, and a Bers slice